Let’s start by playing a game. I roll a dice and pay you in pounds the number that appears on it. How much would you be prepared to pay to play? If you pay £1 you cannot lose, and if you pay £6 you cannot win but at what point do the odds tip from my advantage to yours?

It was the correspondence between two of the greats of mathematics, Fermat and Pascal, that led to the discovery that you could apply mathematics to analyse games of chance. Previous generations had not dreamt of such a connection. Mathematics is a subject of certainties and truths. How can it apply to the analysis of chance and randomness? The mathematics of chance crystallised with the publication by Swiss mathematician Jakob Bernoulli of Ars Conjectandi, or The Art of Conjecture. It is here that you find the formula for the fair price that you should pay for any game.

Suppose there are N possible outcomes (in our dice game, N=6). You win W (1) pounds if outcome 1 occurs (ie, £1 for a roll of 1). This happens with probability P (1) (in this case, 1/6). Similarly, outcome 2 occurs with probability P (2) in which case you win W (2) pounds (in our game, £2 for a roll of 2, again with a probability of 1/6). On average, a game earns you W (1) x P (1) +…+W (N) x P (N) pounds each time you play, which in our dice game equals £1 x 1/6 + £2 x 1/6… + £6 x 1/6 = £3.50. So if I offered you less than this to play, then you’re going to be the winner in the long run.

The formula seemed sound until Jakob Bernoulli’s cousin Nicolaus, in an almost oedipal act, came up with the following game: I toss a coin. If it lands heads I pay you £2 and the game ends. If it lands tails then I toss again. If the second toss is heads I pay you £4. If it is tails I toss again. Each time I toss, the payout doubles. So if I toss six tails followed by a head I’ll pay you 2 x 2 x 2 x 2 x 2 x 2 x 2 = 2 to the power of 7 = £128. How much would you be prepared to pay to play Nicolaus’s game? Four pounds? Twenty pounds? One hundred pounds?

Well, there’s a 50 per cent chance that you’ll win only £2. After all, the probability that it lands heads on the first toss is 1/2. So P (1) = 1/2 and W (1) = 2. But you’re hoping for a long run of tails followed by a head to get as big a prize as possible. The probability that you get a tail followed by a head will be 1/2 x 1/2 = 1/4. But this time you win £4. So the second outcome has P (2) = 1/4 but W (2) = 4. As you keep going the probabilities get smaller but the payout bigger. For example, six tails followed by a head has a probability of (1/2) to the power of 7 = 1/128 but wins you 2 to the power of 7 = £128.

If you stopped the game after seven tosses then you would lose only if there were seven tails in a row. Using Jakob’s formula the average payout would be W (1) x P (1) +…+ W (7) x P (7) = (1/2 x 2) + (1/4 x 4) +…+ (1/128 x 128) = 1 + 1 +…+ 1 = £7. It is worth playing the game, therefore, if anyone offers you less than £7 to play.

But here is the sting. Nicolaus is prepared to play the game indefinitely until a head appears. You’re a winner every time. So how much will you pay to play the game?

There are infinitely many options now. The formula says that the average payout will be 1 + 1 + 1 +… namely infinity pounds! If anyone offers to play this game with you, it’s worth playing whatever the cost to play. In the long run the maths says that you will come out on top. But why is it that most of us wouldn’t play the game for anything more than about £10?

It’s called the St Petersburg Paradox after Nicolaus’s cousin Daniel who, while working at the Imperial Academy of Sciences in St Petersburg, came up with the first explanation of why no rational person would pay any price to play the game. The answer is what any billionaire will tell you. The first million you earn is worth so much more than the second million. You shouldn’t put in the formula the exact amount you win but what that prize is worth to you. In this way the price to play this game will vary according to how you value the outcomes. Daniel’s resolution goes far beyond just the curiosity of a mathematical game: it is essentially the foundation of modern economics.

Go to www.mathematik.com/ Petersburg/Petersburg.html for an online simulation of the game

Conundrum

If you could play a game a second, how long would it take to play 2 to the power of 60 games? This is the number of games you might expect to play to break even in the Petersburg game if the entry price was £60.

Answer

More than 36 billion years. The universe is at most 14 billion years old — another explanation for why most people wouldn’t pay an arbitrary price to play the game.

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